Let \(({\mathbb {X}},{\mathcal {X}})\)  be a Borel space equipped with a
\(\sigma \) -finite measure \(\lambda \)  and define
the set ring \({\mathcal {X}}_0:=\lbrace B\in {\mathcal {X}}:\lambda (B)<\infty \rbrace \) .
Let \({\mathbf {N}}\equiv {\mathbf {N}}({\mathbb {X}})\)  be the space of all measures on \({\mathbb {X}}\) 
which are \({\mathbb {N}}_0\) -valued on \({\mathcal {X}}_0\)  (where \({\mathbb {N}}_0:={\mathbb {N}}\cup \lbrace 0\rbrace =\lbrace 0,1,2,\ldots \rbrace \) ) and
let \({\mathcal {N}}\)  denote the smallest \(\sigma \) -field
such that \(\mu \mapsto \mu (B)\)  is measurable for
all \(B\in {\mathcal {X}}\) . A point process is a random
element \(\eta \)  of \({\mathbf {N}}\) , defined over some fixed
probability space \((\Omega ,{\mathcal {F}},\mathbb {P})\) .
The intensity measure of \(\eta \)  is the
measure \(\mathbb {E}[\eta ]\)  defined by \(\mathbb {E}[\eta ](B):=\mathbb {E}\eta (B)\) , \(B\in {\mathcal {X}}\) .
For any \(\sigma \) -finite measure \(\nu \)  on \({\mathbb {X}}\)  we let
\(\Pi _\nu \)  denote the distribution of a Poisson process
(see e.g. [1]}) on \({\mathbb {X}}\) 
with intensity measure \(\nu \) . Of particular interest for
us are the distributions \(\Pi _{\alpha \lambda _B}\) , where \(\alpha \ge 0\) 
and \(\lambda _B:=\lambda (B\cap \cdot )\)  is the restriction of
\(\lambda \)  to some \(B\in {\mathcal {X}}_0\) . We use the latter notation for any
measure on \({\mathbb {X}}\) . For each \(B\in {\mathcal {X}}\)  we define
\({\mathbf {N}}_B:=\lbrace \mu \in {\mathbf {N}}:\mu (B^c)=0\rbrace =\lbrace \mu _B:\mu \in {\mathbf {N}}\rbrace \) .
For \(\mu \in {\mathbf {N}}\)  we write \(x \in \mu \)  if \(\mu (\lbrace x\rbrace )>0\) .
If \(\mu (\lbrace x\rbrace )\in \lbrace 0,1\rbrace \)  for all \(x\in {\mathbb {X}}\)  then
\(\mu \)  is called simple.
